Simplify the following expression and state the conditions under which the simplification is valid. You can assume that $n \neq 0$. $y = \dfrac{2n - 6}{-n^2 + 11n - 24} \times \dfrac{n^2 - 10n + 16}{5n + 10} $
Solution: First factor out any common factors. $y = \dfrac{2(n - 3)}{-(n^2 - 11n + 24)} \times \dfrac{n^2 - 10n + 16}{5(n + 2)} $ Then factor the quadratic expressions. $y = \dfrac {2(n - 3)} {-(n - 8)(n - 3)} \times \dfrac {(n - 8)(n - 2)} {5(n + 2)} $ Then multiply the two numerators and multiply the two denominators. $y = \dfrac {2(n - 3) \times (n - 8)(n - 2) } { -(n - 8)(n - 3) \times 5(n + 2)} $ $y = \dfrac {2(n - 8)(n - 2)(n - 3)} {-5(n - 8)(n - 3)(n + 2)} $ Notice that $(n - 8)$ and $(n - 3)$ appear in both the numerator and denominator so we can cancel them. $y = \dfrac {2\cancel{(n - 8)}(n - 2)(n - 3)} {-5\cancel{(n - 8)}(n - 3)(n + 2)} $ We are dividing by $n - 8$ , so $n - 8 \neq 0$ Therefore, $n \neq 8$ $y = \dfrac {2\cancel{(n - 8)}(n - 2)\cancel{(n - 3)}} {-5\cancel{(n - 8)}\cancel{(n - 3)}(n + 2)} $ We are dividing by $n - 3$ , so $n - 3 \neq 0$ Therefore, $n \neq 3$ $y = \dfrac {2(n - 2)} {-5(n + 2)} $ $ y = \dfrac{-2(n - 2)}{5(n + 2)}; n \neq 8; n \neq 3 $